Charge density wave states in tantalum dichalcogenides

Using density functional theory, we explore a range of charge density wave states (CDWs) in tantalum-based transition-metal dichalcogenide monolayers. The high-symmetry states of the $1H$ phases of $\mathrm{Ta}{X}_2$ ($X$ = S, Se, Te) are lower in total energy compared to the $1T$ variants, while the $1T$ phases exhibit a much stronger tendency for CDW formation. The stability of several CDWs is found to be stronger as the chalcogenide is changed in the sequence (S, Se, Te), with the tellurium-based systems exhibiting several CDWs with binding energy per formula unit in the range of $100\mathrm{meV}$. These $1T$ CDW phases are lower in energy than the corresponding $1H$ CDW phases. The diversity of CDWs exhibited by these materials suggests that many “hidden” states may occur on ultrafast excitation or photodoping. Changes in electronic structure across the $\mathrm{Ta}{X}_2$ series are also elucidated.

Figure 1

The Star of David structure (SoD). Labeled are ${\vec{a}}^{\prime }$ and ${\vec{b}}^{\prime }$, the new lattice vectors of the $\sqrt{13}\times \sqrt{13}$ supercell. Also shown are $\vec{a}$ and $\vec{b}$, the primitive lattice vectors. The three unique sites of Ta atoms in the SoD structure (A, B, and C) are labeled. The CDW contraction involves a shortening of the A-B and A-C distances along with a rotation of $13.6^{\circ }$ relative to the primitive lattice vectors. Charge accumulation is more pronounced on the A site than the B sites.

Figure 2

The $1T$ (left) and $1H$ (right) primitive structures of $\mathrm{Ta}{X}_2$. The $1T$ structure has octahedral symmetry (top left), while the $1H$ structure displays trigonal prismatic symmetry (top right). The bottom illustrations are side views of the first plane of atoms.

Figure 3

The distortion of Ta atoms in the $3\times 3$ CDW in $1T-{\mathrm{TaTe}}_2$. The superlattice vectors (${\vec{a}}^{\prime }$ and ${\vec{b}}^{\prime }$) are pictured, and the four sites (A, B, C, and D) are labeled. Charge accumulates around the A site with contraction of the A-B, A-C, and B-C interatomic distances. The C-D distance remains similar to the high-symmetry structure, but the B-D and C-C distances are larger than in the high-symmetry case.

Figure 4

Side views of the high symmetry (top) and $3\times 3$ CDW (bottom) in 1T-${\mathrm{TaTe}}_2$. In the bottom plot, the Ta sites are labeled according to the definitions in Fig. 3. This highlights the buckling of the Te sites and the increase in the Te-Te layer thickness.

Figure 5

The distortions of Ta atoms in the four-atom striped CDW occurring in $1T-{\mathrm{TaSe}}_2$ and $1T-{\mathrm{TaS}}_2$. Only the supercell lattice vectors (${\vec{a}}^{\prime }$ and ${\vec{b}}^{\prime }$) are given, and the four unique sites (A, B, C, and D) are labeled. This CDW involves a shortening of the B-C distances, while the A-B and C-D distances remain nearly the same as in the high-symmetry structure.

Figure 6

The distortion of Ta atoms in the four-atom striped CDW occurring in $1T-{\mathrm{TaTe}}_2$. Only the supercell lattice vectors (${\vec{a}}^{\prime }$ and ${\vec{b}}^{\prime }$) are pictured, and the four unique sites (A, B, C, and D) are labeled. This CDW involves a shortening of the A-B, B-C, and C-D distances; however, the B-C contraction is less than the A-B and C-D contractions. Combined with displacement of Ta atoms in the $y$ direction perpendicular to the ${\vec{a}}^{\prime }$ lattice vector (upwards in the figure), this results in a two-stripe internal structure within the $4\times 1$ CDW.

Figure 7

Band structures of the primitive unit cells for the $1T-\mathrm{Ta}{X}_2$ materials for (a) S, (b) Se, and (c) Te. The black lines are the standard GGA-PBE calculation with spin polarization, while the orange lines include spin-orbit coupling.

Figure 8

Band structures of the high-symmetry structure (no CDW) for the $1H-\mathrm{Ta}{X}_2$ materials for (a) S, (b) Se, and (c) Te. The solid (dashed) lines represent the majority (minority) spin for spin-degenerate bands. The black lines are the standard GGA-PBE calculation with spin polarization, while the red lines include spin-orbit coupling.

Figure 9

Plot of the energy per formula unit for $1T-{\mathrm{TaS}}_2$ in a $\sqrt{13}\times \sqrt{13}$ unit cell for high-symmetry and CDW phases. Energies are plotted relative to the high-symmetry ground-state energy. Zero strain refers to the lattice parameters of the lowest-energy CDW structure, which differ slightly from the lattice parameters of the high-symmetry ground state. The lowest energy for the high-symmetry unit cell occurs at $0.41\%$ compressive strain.

Figure 10

Decomposed density of states for the SoD structures in $1T-{\mathrm{TaS}}_2$ and $1T-{\mathrm{TaSe}}_2$. The solid black line is the total density of states about the Fermi energy, while the dashed red and dot-dashed green lines are the contributions from the Ta $d$ orbitals and the chalcogen $p$ orbitals. In $1T-{\mathrm{TaS}}_2$ the contribution at the Fermi energy is mostly from the Ta $d$ orbitals, while in $1T-{\mathrm{TaSe}}_2$ there is a significant contribution from the Se $p$ orbitals. This leads to the exchange splitting of $\sim 0.25\mathrm{eV}$ in $1T-{\mathrm{TaS}}_2$.

Figure 11

Band structure for the SoD structure in (a) $1T-{\mathrm{TaS}}_2$, (b) $1T-{\mathrm{TaSe}}_2$, and (c) $1T-{\mathrm{TaTe}}_2$. where solid (dashed) lines represent majority (minority) spin in cases where this is net spin polarization. The energy is plotted with respect to the Fermi energy.

Figure 12

Band structure for the $4\times 1$ CDW structure in (a) $1T-{\mathrm{TaS}}_2$, (b) $1T-{\mathrm{TaSe}}_2$, and (c) $1T-{\mathrm{TaTe}}_2$, where solid (dashed) lines represent majority (minority) spin. The energy is plotted with respect to the Fermi energy.